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Item 1. Formula Sheet-PC Math 12. Below is the formula sheet for this course, as found in the course intro.
This item is a formula sheet/reference item. No calculation is required. Use the formula sheet as needed throughout the quiz. Answer: N/N
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Item 2. The graph of a periodic trigonometric function of the form \(y=a\cos(x-c)+d\) is graphed below. Determine the smallest value of \(c\). [Graph will be inserted later.]
In the function \(y=a\cos(x-c)+d\), the value \(c\) represents the phase shift. The maximum point of the cosine graph helps identify the phase shift. From the graph, the maximum occurs at \(x=\frac{\pi}{2}\). Therefore, the smallest value of \(c\) is \(\frac{\pi}{2}\). Answer: D
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Item 3. The graph of a periodic trigonometric function of the form \(y=a\sin(x-c)+d\) is graphed below. Determine the value of \(d\). [Graph will be inserted later.]
In the function \(y=a\sin(x-c)+d\), the value \(d\) represents the vertical translation, also called the equilibrium line. The equilibrium line can be found using the formula: \(d=\frac{\text{max}+\text{min}}{2}\) From the graph, the maximum value is \(1\), and the minimum value is \(-3\). \(d=\frac{1+(-3)}{2}\) \(d=\frac{-2}{2}\) \(d=-1\) Therefore, the value of \(d\) is \(-1\). Answer: A
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Item 4. Of the following equations, which have undergone a horizontal expansion from their original equation: \(y=\sin x\) or \(y=\cos x\)? I. \(y=\cos 3x\) II. \(y=\sin\left(\frac{3}{2}x\right)\) III. \(y=\cos\left(\frac{1}{4}x\right)\) IV. \(y=\sin\left(\frac{2}{3}x\right)\) V. \(y=\cos\left(\frac{4}{3}x\right)\)
For \(y=\sin bx\) or \(y=\cos bx\), a horizontal expansion occurs when \(0<|b|<1\). I. \(y=\cos 3x\): \(b=3\), so this is not a horizontal expansion. II. \(y=\sin\left(\frac{3}{2}x\right)\): \(b=\frac{3}{2}\), so this is not a horizontal expansion. III. \(y=\cos\left(\frac{1}{4}x\right)\): \(b=\frac{1}{4}\), so \(0<|b|<1\). This is a horizontal expansion. IV. \(y=\sin\left(\frac{2}{3}x\right)\): \(b=\frac{2}{3}\), so \(0<|b|<1\). This is a horizontal expansion. V. \(y=\cos\left(\frac{4}{3}x\right)\): \(b=\frac{4}{3}\), so this is not a horizontal expansion. Therefore, III and IV have undergone a horizontal expansion. Answer: D
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Item 5. For the function \(f(x)=2\cos 8\left(x+\frac{\pi}{3}\right)+1\), the period and the phase shift respectively are:
For a cosine function, the period is \(\frac{2\pi}{b}\). In \(f(x)=2\cos 8\left(x+\frac{\pi}{3}\right)+1\), the value of \(b\) is \(8\). \(\text{period}=\frac{2\pi}{8}\) \(\text{period}=\frac{\pi}{4}\) The expression \(x+\frac{\pi}{3}\) means the graph shifts left by \(\frac{\pi}{3}\). Therefore, the period is \(\frac{\pi}{4}\), and the phase shift is \(\frac{\pi}{3}\) left. Answer: D
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Item 6. A mass is supported by a spring so that it rests \(15\text{ cm}\) above a table top. The mass is pulled down to a height of \(5\text{ cm}\) above the table top and released at time \(t=0\). It takes \(0.6\) seconds for the mass to reach a maximum height of \(25\text{ cm}\) above the table top. As the mass moves up and down, its height \(h\), above the table top, is approximated by a sinusoidal function of the elapsed time \(t\), for a short period of time. Determine which of the following is the cosine equation that gives the height, \(h\), as a function of time, \(t\).
The minimum height is \(5\text{ cm}\), and the maximum height is \(25\text{ cm}\). The amplitude can be found by: \(\text{amp}=\frac{\text{max}-\text{min}}{2}\) \(\text{amp}=\frac{25-5}{2}=10\) The vertical displacement can be found by: \(d=\frac{\text{max}+\text{min}}{2}\) \(d=\frac{25+5}{2}=15\) The time from the minimum to the maximum is \(0.6\) seconds, which is half of the period. \(\text{period}=2(0.6)=1.2\) The \(b\)-value is: \(b=\frac{2\pi}{\text{period}}=\frac{2\pi}{1.2}\) Since the mass begins at the minimum height when \(t=0\), the cosine curve must be vertically reflected. Therefore, the equation is \(h=-10\cos\left(\frac{2\pi t}{1.2}\right)+15\). Answer: A
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Item 7. What is the period of the function \(y=2\tan\left(\frac{x}{2}+3\right)\)?
The period of a tangent function is \(\frac{\pi}{b}\). First factor out the coefficient of \(x\) inside the tangent function. \(y=2\tan\left(\frac{x}{2}+3\right)\) \(y=2\tan\left(\frac{1}{2}(x+6)\right)\) So \(b=\frac{1}{2}\). \(\text{period}=\frac{\pi}{b}\) \(\text{period}=\frac{\pi}{\frac{1}{2}}\) \(\text{period}=2\pi\) Therefore, the period is \(2\pi\). Answer: A
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